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Classical mechanics (find trajectory and kinetic energy)

  1. Dec 9, 2015 #1
    1. The problem statement, all variables and given/known data
    Given the force ## \vec{ F }(x) = (-12x + 6) \vec{i} ## ; find kinetic energy ## T## at the point ##x=2## and trajectory of a particle ## \vec{r}(t) ##, given that ## \vec{r}(t=0)=\vec{0}## and ##\dot{\vec{r}}(t=0)=\vec{0}## .

    3. The attempt at a solution

    Since ##\nabla\times \vec{F} = \vec{0}## we can find potential function ## \vec{F}(x) = -\nabla U(x)## , therefore ## U(x)= 6x²-6x + C ## . From here I am not sure how to proceed.
    Second idea is to solve differential equation but I'm not sure how to do it because force is a function of ## x## and not ## t ## .
    $$ m\ddot{x}=-12x+6$$ $$\ddot{x}+\frac{1}{m}12x-\frac{1}{m}6=0$$ let's suppose ##m=1## for simplicity, then we get $$ \ddot{x}+12x-6=0$$
     
  2. jcsd
  3. Dec 11, 2015 #2

    haruspex

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    With a simple change of variable you can eliminate the constant 6 from the differential equation and obtain a very well known form. Think springs.
    By the way, this belongs in Introductory Physics homework.
     
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